Question: Simplify; express your answer in exponential form. Assume $p\neq 0, x\neq 0$. $\dfrac{{(p^{2}x^{-1})^{2}}}{{(p^{-2}x^{-4})^{5}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{2}x^{-1})^{2} = (p^{2})^{2}(x^{-1})^{2}}$ On the left, we have ${p^{2}}$ to the exponent ${2}$ . Now ${2 \times 2 = 4}$ , so ${(p^{2})^{2} = p^{4}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{2}x^{-1})^{2}}}{{(p^{-2}x^{-4})^{5}}} = \dfrac{{p^{4}x^{-2}}}{{p^{-10}x^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{4}x^{-2}}}{{p^{-10}x^{-20}}} = \dfrac{{p^{4}}}{{p^{-10}}} \cdot \dfrac{{x^{-2}}}{{x^{-20}}} = p^{{4} - {(-10)}} \cdot x^{{-2} - {(-20)}} = p^{14}x^{18}$